answer：Okay, so I have this problem here: I need to find how many positive three-digit integers satisfy three different remainder conditions. Specifically, the number should have a remainder of 2 when divided by 6, a remainder of 5 when divided by 9, and a remainder of 7 when divided by 11. The options are from A to E, with E being 5. Hmm, let me think about how to approach this.First, I remember that when dealing with problems that involve remainders and multiple divisors, the Chinese Remainder Theorem (CRT) is usually helpful. But I'm not too familiar with all the details, so maybe I can break it down step by step.Let me denote the three-digit number as N. So, N is between 100 and 999 inclusive. Now, the conditions given are:1. N ≡ 2 mod 62. N ≡ 5 mod 93. N ≡ 7 mod 11I need to find all such N in the three-digit range. Let me tackle these one by one.Starting with the first condition: N ≡ 2 mod 6. That means when N is divided by 6, the remainder is 2. So, N can be written as 6k + 2, where k is some integer. Since N is a three-digit number, 100 ≤ 6k + 2 ≤ 999. Let me solve for k:100 ≤ 6k + 2 ≤ 999 Subtract 2: 98 ≤ 6k ≤ 997 Divide by 6: 98/6 ≈ 16.333 ≤ k ≤ 997/6 ≈ 166.166Since k must be an integer, k ranges from 17 to 166. So, there are 166 - 17 + 1 = 150 possible values for k here. But I don't know if all these will satisfy the other two conditions, so I need to check the other congruences.Moving on to the second condition: N ≡ 5 mod 9. So, N can be written as 9m + 5, where m is an integer. Similarly, since N is three-digit, 100 ≤ 9m + 5 ≤ 999. Let me solve for m:100 ≤ 9m + 5 ≤ 999 Subtract 5: 95 ≤ 9m ≤ 994 Divide by 9: 95/9 ≈ 10.555 ≤ m ≤ 994/9 ≈ 110.444So, m ranges from 11 to 110, giving 110 - 11 + 1 = 100 possible values. Again, not all of these will satisfy the first and third conditions.Third condition: N ≡ 7 mod 11. So, N can be written as 11n + 7, where n is an integer. For three-digit numbers:100 ≤ 11n + 7 ≤ 999 Subtract 7: 93 ≤ 11n ≤ 992 Divide by 11: 93/11 ≈ 8.454 ≤ n ≤ 992/11 ≈ 90.181So, n ranges from 9 to 90, giving 90 - 9 + 1 = 82 possible values.Now, since N has to satisfy all three conditions simultaneously, I need to find numbers that are common to all three sequences: 6k + 2, 9m + 5, and 11n + 7. This is where the Chinese Remainder Theorem comes into play, I think.Let me try to solve the system of congruences step by step.First, let's combine the first two congruences:1. N ≡ 2 mod 62. N ≡ 5 mod 9I can write N as 6k + 2. Substitute this into the second equation:6k + 2 ≡ 5 mod 9 So, 6k ≡ 3 mod 9 Simplify: 6k ≡ 3 mod 9Divide both sides by 3: 2k ≡ 1 mod 3 So, 2k ≡ 1 mod 3To solve for k, I can multiply both sides by the modular inverse of 2 mod 3. The inverse of 2 mod 3 is 2, since 2*2=4≡1 mod3.So, k ≡ 2*1 ≡ 2 mod3 Thus, k = 3t + 2, where t is an integer.Therefore, N = 6k + 2 = 6*(3t + 2) + 2 = 18t + 12 + 2 = 18t + 14So, N ≡ 14 mod18. So, N can be written as 18t +14.Now, let's incorporate the third condition: N ≡7 mod11.So, 18t +14 ≡7 mod11 Let me compute 18 mod11 and 14 mod11 first.18 divided by 11 is 1 with remainder 7, so 18 ≡7 mod11 14 divided by 11 is 1 with remainder 3, so 14≡3 mod11Thus, the equation becomes:7t + 3 ≡7 mod11 Subtract 3: 7t ≡4 mod11Now, I need to solve for t: 7t ≡4 mod11To find t, I need the inverse of 7 mod11. Let me find an integer x such that 7x ≡1 mod11.Testing x=1: 7*1=7≡7≠1 x=2:14≡3 x=3:21≡10 x=4:28≡6 x=5:35≡2 x=8:56≡1 (since 55 is 5*11, so 56≡1)Wait, 7*8=56≡1 mod11, so inverse of 7 mod11 is 8.Therefore, multiply both sides by 8:t ≡4*8 mod11 t≡32 mod11 32 divided by 11 is 2 with remainder 10, so t≡10 mod11Therefore, t=11s +10, where s is an integer.Thus, N=18t +14=18*(11s +10)+14=198s +180 +14=198s +194So, N=198s +194Now, N must be a three-digit number, so 100 ≤198s +194 ≤999Let me solve for s:100 ≤198s +194 ≤999 Subtract 194: 100 -194= -94 ≤198s ≤999 -194=805 So, -94 ≤198s ≤805Divide by 198:-94/198 ≈-0.474 ≤s ≤805/198≈4.065Since s must be an integer, s can be 0,1,2,3,4.Let me compute N for each s:s=0: N=198*0 +194=194 s=1:198 +194=392 s=2:396 +194=590 s=3:594 +194=788 s=4:792 +194=986So, the possible Ns are 194, 392, 590, 788, 986.Now, let me check if these numbers satisfy all three original conditions.First, check N=194:194 divided by6: 6*32=192, remainder 2. Good. 194 divided by9: 9*21=189, remainder5. Good. 194 divided by11:11*17=187, remainder7. Good.Next, N=392:392/6=65*6=390, remainder2. Good. 392/9=43*9=387, remainder5. Good. 392/11=35*11=385, remainder7. Good.N=590:590/6=98*6=588, remainder2. Good. 590/9=65*9=585, remainder5. Good. 590/11=53*11=583, remainder7. Good.N=788:788/6=131*6=786, remainder2. Good. 788/9=87*9=783, remainder5. Good. 788/11=71*11=781, remainder7. Good.N=986:986/6=164*6=984, remainder2. Good. 986/9=109*9=981, remainder5. Good. 986/11=89*11=979, remainder7. Good.All five numbers satisfy all three conditions. Wait, but the options only go up to E)5. So, does that mean the answer is 5? But the options are (A)1, (B)2, (C)3, (D)4, (E)5.Wait, but hold on. The problem says "positive three-digit integers." So, 194 is a three-digit number, as are 392, 590, 788, and 986. So, that's five numbers. But the options don't have E as 5? Wait, actually, looking back, the options are:(A)1, (B)2, (C)3, (D)4, (E)5.So, E is 5. So, is the answer E)5?Wait, but hold on. Let me double-check my calculations because sometimes when you solve congruences, especially with multiple moduli, you might get overlapping solutions or something.Wait, so I found that N=198s +194, and s=0 gives N=194, which is a three-digit number. s=1 gives 392, s=2 gives 590, s=3 gives 788, s=4 gives 986. s=5 would give 198*5 +194=990 +194=1184, which is four digits, so that's out of range. So, s=0 to s=4 gives five numbers.But wait, the options only go up to E)5, so 5 is an option. So, is the answer E)5? But the options are given as (A)1, (B)2, (C)3, (D)4, (E)5.Wait, but let me check if all these numbers are indeed three-digit numbers. 194 is three-digit, yes. 392, 590, 788, 986—all are three-digit. So, that's five numbers. So, the answer should be 5, which is option E.But wait, looking back at the problem statement, it says "positive three-digit integers." So, 194 is positive and three-digit, so that's fine. So, five numbers in total.Wait, but the answer choices only go up to E)5, so 5 is an option. So, is the answer E)5? Hmm, but in my initial solving, I thought maybe 4, but no, it's 5.Wait, but let me check if I made a mistake in the modulus somewhere.Wait, when I solved 7t ≡4 mod11, I found that t≡10 mod11, so t=11s +10. Then, N=198s +194. So, s=0 gives 194, which is correct. s=1 gives 392, s=2 gives 590, s=3 gives 788, s=4 gives 986, and s=5 gives 1184, which is four digits. So, s=0 to s=4 gives five numbers.Wait, but let me check if 194 is indeed a three-digit number. Yes, 194 is between 100 and 999. So, all five numbers are valid.Wait, but the answer choices don't have E as 5? Wait, no, they do. The options are:(A)1, (B)2, (C)3, (D)4, (E)5.So, E is 5, which is the correct answer. So, the answer is E)5.Wait, but hold on. I just want to make sure that all these numbers actually satisfy all three conditions.Let me pick one, say 590.590 divided by6: 6*98=588, remainder2. Good.590 divided by9: 9*65=585, remainder5. Good.590 divided by11: 11*53=583, remainder7. Good.Another one: 788.788 divided by6: 6*131=786, remainder2. Good.788 divided by9: 9*87=783, remainder5. Good.788 divided by11: 11*71=781, remainder7. Good.And 986:986 divided by6: 6*164=984, remainder2. Good.986 divided by9: 9*109=981, remainder5. Good.986 divided by11: 11*89=979, remainder7. Good.So, all five numbers satisfy the conditions. Therefore, the answer should be 5, which is option E.Wait, but the initial problem didn't specify whether the answer is E or not. Wait, the options are given as (A)1, (B)2, (C)3, (D)4, (E)5. So, yes, 5 is an option, so the answer is E.Wait, but hold on a second. I think I might have made a mistake in the modulus when solving for t. Let me double-check that step.We had N=18t +14, and N≡7 mod11.So, 18t +14 ≡7 mod11.18 mod11 is 7, and 14 mod11 is 3.So, 7t +3 ≡7 mod11.Subtract 3: 7t ≡4 mod11.Then, inverse of 7 mod11 is 8, since 7*8=56≡1 mod11.So, t≡4*8=32≡10 mod11.So, t=11s +10.So, N=18*(11s +10)+14=198s +180 +14=198s +194.So, that seems correct.Then, solving for s:198s +194 ≥100: 198s ≥-94. Since s is integer, s≥0.198s +194 ≤999: 198s ≤805. So, s ≤805/198≈4.065, so s=0,1,2,3,4.So, s=0:194, s=1:392, s=2:590, s=3:788, s=4:986.So, five numbers. So, the answer is 5, which is E.Wait, but the initial problem didn't specify that the answer is E, but in the options, E is 5, so that's correct.Wait, but hold on. Maybe I made a mistake in the initial step when combining the first two congruences.Let me go back.First congruence: N≡2 mod6.Second: N≡5 mod9.Express N as 6k +2.Then, 6k +2 ≡5 mod9.So, 6k ≡3 mod9.Divide both sides by 3: 2k≡1 mod3.Inverse of 2 mod3 is 2, since 2*2=4≡1 mod3.So, k≡2*1=2 mod3.Thus, k=3t +2.So, N=6*(3t +2)+2=18t +14.So, N≡14 mod18.So, N=18t +14.Then, third condition: N≡7 mod11.So, 18t +14≡7 mod11.18 mod11=7, 14 mod11=3.So, 7t +3≡7 mod11.7t≡4 mod11.Inverse of 7 mod11 is 8, so t≡4*8=32≡10 mod11.Thus, t=11s +10.So, N=18*(11s +10)+14=198s +180 +14=198s +194.So, N=198s +194.Then, s=0:194, s=1:392, s=2:590, s=3:788, s=4:986.All are three-digit numbers, so five numbers.Therefore, the answer is 5, which is option E.Wait, but the problem is from an exam, and sometimes the answer might not be as straightforward. Let me check if I missed something.Wait, perhaps I made a mistake in calculating the modulus when combining the first two congruences. Let me check that again.N≡2 mod6 and N≡5 mod9.Express N as 6k +2.Then, 6k +2 ≡5 mod9.So, 6k ≡3 mod9.Divide both sides by 3: 2k≡1 mod3.So, 2k≡1 mod3.Multiplicative inverse of 2 mod3 is 2, since 2*2=4≡1 mod3.Thus, k≡2*1=2 mod3.So, k=3t +2.Therefore, N=6*(3t +2)+2=18t +14.Yes, that seems correct.Then, N=18t +14.Then, N≡7 mod11.So, 18t +14≡7 mod11.18 mod11=7, 14 mod11=3.So, 7t +3≡7 mod11.Subtract 3:7t≡4 mod11.Inverse of 7 mod11 is 8, so t≡4*8=32≡10 mod11.Thus, t=11s +10.So, N=18*(11s +10)+14=198s +194.Yes, that's correct.So, s=0:194, s=1:392, s=2:590, s=3:788, s=4:986.All are three-digit numbers, so five numbers.Therefore, the answer is 5, which is option E.Wait, but the problem is from an exam, and sometimes the answer might not be as straightforward. Let me check if I missed something.Wait, perhaps I made a mistake in the modulus when combining the first two congruences. Let me check that again.N≡2 mod6 and N≡5 mod9.Express N as 6k +2.Then, 6k +2 ≡5 mod9.So, 6k ≡3 mod9.Divide both sides by 3: 2k≡1 mod3.So, 2k≡1 mod3.Multiplicative inverse of 2 mod3 is 2, since 2*2=4≡1 mod3.Thus, k≡2*1=2 mod3.So, k=3t +2.Therefore, N=6*(3t +2)+2=18t +14.Yes, that seems correct.Then, N=18t +14.Then, N≡7 mod11.So, 18t +14≡7 mod11.18 mod11=7, 14 mod11=3.So, 7t +3≡7 mod11.Subtract 3:7t≡4 mod11.Inverse of 7 mod11 is 8, so t≡4*8=32≡10 mod11.Thus, t=11s +10.So, N=18*(11s +10)+14=198s +194.Yes, that's correct.So, s=0:194, s=1:392, s=2:590, s=3:788, s=4:986.All are three-digit numbers, so five numbers.Therefore, the answer is 5, which is option E.Wait, but hold on. Let me check if 194 is indeed a three-digit number. Yes, it is. So, all five numbers are valid.Therefore, the answer is E)5.**Final Answer**boxed{E}

answer：Alright, so I have this problem here: Given the point (-1, 2), two tangent lines are drawn to the parabola C: y² = 4x. I need to find the slope of the line connecting the tangent points and the focus of the parabola. Hmm, okay. Let me try to figure this out step by step.First, I remember that the standard form of a parabola that opens to the right is y² = 4ax, where 'a' is the distance from the vertex to the focus. In this case, comparing to the given equation y² = 4x, it seems like 4a = 4, so a = 1. That means the focus of the parabola is at (a, 0) which is (1, 0). Got that down.Now, the point given is (-1, 2). I need to find the equations of the two tangent lines from this point to the parabola. I remember that the equation of a tangent to a parabola y² = 4ax can be written in the form y = mx + a/m, where m is the slope of the tangent. But wait, is that correct? Let me double-check.Yes, for the parabola y² = 4ax, the tangent in slope form is y = mx + a/m. So, in this case, since a = 1, the equation becomes y = mx + 1/m. Okay, so that's the general form of the tangent.But we have an external point (-1, 2) from which two tangents can be drawn to the parabola. So, the point (-1, 2) must satisfy the tangent equation. Let me substitute x = -1 and y = 2 into the tangent equation.So, substituting, we get 2 = m*(-1) + 1/m. Let me write that down:2 = -m + (1/m)Hmm, that's an equation in terms of m. Let me rearrange it to solve for m.Multiply both sides by m to eliminate the denominator:2m = -m² + 1Bring all terms to one side:m² + 2m - 1 = 0Wait, is that correct? Let me check:Starting from 2 = -m + 1/mMultiply both sides by m: 2m = -m² + 1Bring all terms to left: m² + 2m - 1 = 0Yes, that seems right. So, quadratic equation in m: m² + 2m - 1 = 0Let me solve for m using the quadratic formula. The quadratic is m² + 2m - 1 = 0, so a = 1, b = 2, c = -1.Discriminant D = b² - 4ac = (2)² - 4*1*(-1) = 4 + 4 = 8So, m = [-b ± sqrt(D)] / (2a) = [-2 ± sqrt(8)] / 2Simplify sqrt(8) as 2*sqrt(2), so:m = [-2 ± 2*sqrt(2)] / 2 = [-1 ± sqrt(2)]So, the two slopes are m1 = -1 + sqrt(2) and m2 = -1 - sqrt(2). Okay, so that gives us the slopes of the two tangent lines from the point (-1, 2) to the parabola.Now, I need to find the points of tangency on the parabola. Once I have those points, I can find the line connecting them and then find its slope. Alternatively, maybe I can find the equation of the chord of contact and then find the midpoint or something? Hmm, not sure. Let me think.Wait, the chord of contact from an external point to a parabola is the line joining the points of contact of the two tangents. So, maybe if I find the equation of the chord of contact, I can find the two points where it intersects the parabola, which would be the tangent points.But I think another approach is, since I have the slopes of the tangents, I can find the points of contact.Given that the tangent equation is y = mx + 1/m, and the parabola is y² = 4x.To find the point of contact, I can solve these two equations together.Let me substitute y from the tangent into the parabola equation:(mx + 1/m)² = 4xExpanding the left side:m²x² + 2*(mx)*(1/m) + (1/m²) = 4xSimplify:m²x² + 2x + (1/m²) = 4xBring all terms to one side:m²x² + 2x + (1/m²) - 4x = 0Simplify:m²x² - 2x + (1/m²) = 0This is a quadratic in x. Since the line is tangent to the parabola, this quadratic should have exactly one solution, meaning discriminant is zero. But wait, I already know that, so maybe I can find the x-coordinate of the point of contact.Alternatively, maybe I can use the formula for the point of contact. For a parabola y² = 4ax, the point of contact of the tangent y = mx + a/m is (a/m², 2a/m). So, in this case, a = 1, so the point is (1/m², 2/m). That seems useful.Yes, that's a formula I remember. So, for each slope m, the point of contact is (1/m², 2/m). So, for m1 = -1 + sqrt(2), the point is (1/(m1²), 2/m1), and similarly for m2.So, let me compute these points.First, compute for m1 = -1 + sqrt(2):Compute m1²:m1 = -1 + sqrt(2)m1² = (-1 + sqrt(2))² = 1 - 2*sqrt(2) + 2 = 3 - 2*sqrt(2)So, 1/m1² = 1/(3 - 2*sqrt(2)). Let me rationalize the denominator:Multiply numerator and denominator by (3 + 2*sqrt(2)):1/(3 - 2*sqrt(2)) * (3 + 2*sqrt(2))/(3 + 2*sqrt(2)) = (3 + 2*sqrt(2))/(9 - (2*sqrt(2))²) = (3 + 2*sqrt(2))/(9 - 8) = (3 + 2*sqrt(2))/1 = 3 + 2*sqrt(2)So, 1/m1² = 3 + 2*sqrt(2)Similarly, 2/m1 = 2/(-1 + sqrt(2)). Let me rationalize this as well:Multiply numerator and denominator by (-1 - sqrt(2)):2*(-1 - sqrt(2))/((-1 + sqrt(2))*(-1 - sqrt(2))) = (-2 - 2*sqrt(2))/(1 - (sqrt(2))²) = (-2 - 2*sqrt(2))/(1 - 2) = (-2 - 2*sqrt(2))/(-1) = 2 + 2*sqrt(2)So, 2/m1 = 2 + 2*sqrt(2)Therefore, the point of contact for m1 is (3 + 2*sqrt(2), 2 + 2*sqrt(2))Similarly, let's compute for m2 = -1 - sqrt(2):Compute m2²:m2 = -1 - sqrt(2)m2² = (-1 - sqrt(2))² = 1 + 2*sqrt(2) + 2 = 3 + 2*sqrt(2)So, 1/m2² = 1/(3 + 2*sqrt(2)). Rationalizing:Multiply numerator and denominator by (3 - 2*sqrt(2)):1/(3 + 2*sqrt(2)) * (3 - 2*sqrt(2))/(3 - 2*sqrt(2)) = (3 - 2*sqrt(2))/(9 - (2*sqrt(2))²) = (3 - 2*sqrt(2))/(9 - 8) = (3 - 2*sqrt(2))/1 = 3 - 2*sqrt(2)So, 1/m2² = 3 - 2*sqrt(2)Similarly, 2/m2 = 2/(-1 - sqrt(2)). Rationalizing:Multiply numerator and denominator by (-1 + sqrt(2)):2*(-1 + sqrt(2))/((-1 - sqrt(2))*(-1 + sqrt(2))) = (-2 + 2*sqrt(2))/(1 - (sqrt(2))²) = (-2 + 2*sqrt(2))/(1 - 2) = (-2 + 2*sqrt(2))/(-1) = 2 - 2*sqrt(2)So, 2/m2 = 2 - 2*sqrt(2)Therefore, the point of contact for m2 is (3 - 2*sqrt(2), 2 - 2*sqrt(2))So, now I have the two points of tangency:Point P: (3 + 2*sqrt(2), 2 + 2*sqrt(2))Point Q: (3 - 2*sqrt(2), 2 - 2*sqrt(2))Now, I need to find the slope of the line connecting these two points. Wait, but the question says "the slope of the line connecting the tangent points and the focus of the parabola." Hmm, does that mean connecting each tangent point to the focus, or connecting the two tangent points and then considering that line? The wording is a bit unclear.Wait, let me read it again: "The slope of the line connecting the tangent points and the focus of the parabola is _______."Hmm, so "the line connecting the tangent points and the focus." So, perhaps the line that connects both tangent points and the focus? But a line can't connect three points unless they are colinear. Alternatively, maybe it's the line connecting the two tangent points, and then also mentioning the focus? Or perhaps it's the line connecting the two tangent points, and we need to find its slope.Wait, let me think. The focus is at (1, 0). So, if I connect the two tangent points, which are P and Q, and then also connect them to the focus, but the question is about the slope of the line connecting the tangent points and the focus. Hmm, maybe it's the line that connects both tangent points and passes through the focus? But I don't think that's necessarily the case.Alternatively, maybe it's the line connecting the two tangent points, and the focus is another point, but the slope is just between the two tangent points. Hmm, the wording is a bit ambiguous.Wait, let me reread the problem: "The slope of the line connecting the tangent points and the focus of the parabola is _______."Hmm, so it's a single line that connects the tangent points and the focus. So, perhaps the line passes through both tangent points and the focus? But unless the three points are colinear, which I don't think they are, that might not be the case.Alternatively, maybe it's the line connecting the two tangent points, and separately, the focus is another point, but the slope is just of the line connecting the two tangent points. Maybe the wording is a bit off.Wait, let me think again. The problem says: "two tangent lines are drawn to the parabola C: y²=4x. The slope of the line connecting the tangent points and the focus of the parabola is _______."So, perhaps it's the line that connects the two tangent points (i.e., the chord of contact) and also connects to the focus. But that might not make sense because a line can't connect three points unless they are colinear.Alternatively, maybe it's the line connecting each tangent point to the focus, but the slope would be different for each. Hmm, but the question is asking for a single slope.Wait, perhaps it's the line connecting the two tangent points, which is the chord of contact, and then the slope of that line. That makes sense because the chord of contact is a single line, so its slope can be found.Alternatively, maybe it's the line connecting the two tangent points and the focus, but that seems like it's asking for the slope of a line that connects three points, which is not a standard thing.Wait, actually, in the problem statement, it's written as "the line connecting the tangent points and the focus." So, maybe it's a line that connects both tangent points and the focus, but that would require all three points to be on the same line, which may not be the case.Wait, let's check if the focus (1,0) lies on the chord of contact PQ.We have points P: (3 + 2*sqrt(2), 2 + 2*sqrt(2)) and Q: (3 - 2*sqrt(2), 2 - 2*sqrt(2))Let me compute the equation of the line PQ.First, find the slope of PQ.Slope m = (y2 - y1)/(x2 - x1)Compute y2 - y1: [2 - 2*sqrt(2)] - [2 + 2*sqrt(2)] = -4*sqrt(2)Compute x2 - x1: [3 - 2*sqrt(2)] - [3 + 2*sqrt(2)] = -4*sqrt(2)So, slope m = (-4*sqrt(2))/(-4*sqrt(2)) = 1So, the slope of PQ is 1.Now, let's find the equation of PQ. Using point P: (3 + 2*sqrt(2), 2 + 2*sqrt(2))Equation: y - y1 = m(x - x1)So, y - (2 + 2*sqrt(2)) = 1*(x - (3 + 2*sqrt(2)))Simplify:y = x - 3 - 2*sqrt(2) + 2 + 2*sqrt(2)Simplify constants: (-3 + 2) = -1Simplify sqrt terms: (-2*sqrt(2) + 2*sqrt(2)) = 0So, equation: y = x - 1So, the chord of contact PQ has the equation y = x - 1.Now, does the focus (1, 0) lie on this line?Plug x = 1 into y = x - 1: y = 1 - 1 = 0. Yes, it does. So, the focus lies on the chord of contact PQ.Therefore, the line connecting the tangent points (P and Q) and the focus is the same line PQ, which has the equation y = x - 1, and its slope is 1.Therefore, the slope is 1.Wait, that seems straightforward. So, the slope is 1.But let me double-check everything to make sure I didn't make a mistake.First, I found the slopes of the tangents correctly: m1 = -1 + sqrt(2), m2 = -1 - sqrt(2). Then, using the point of contact formula, I found points P and Q correctly.Then, I calculated the slope between P and Q as (y2 - y1)/(x2 - x1). Let me recalculate that:Point P: (3 + 2√2, 2 + 2√2)Point Q: (3 - 2√2, 2 - 2√2)So, y2 - y1 = (2 - 2√2) - (2 + 2√2) = -4√2x2 - x1 = (3 - 2√2) - (3 + 2√2) = -4√2So, slope = (-4√2)/(-4√2) = 1. Correct.Then, equation of PQ: y = x - 1. Focus is (1,0). Plugging in, 0 = 1 - 1 = 0. Correct.Therefore, the slope is indeed 1.Wait, but just to make sure, let me think about another approach. Maybe using parametric equations of the parabola.For parabola y² = 4x, parametric equations are x = at², y = 2at. Here, a = 1, so x = t², y = 2t.The equation of the tangent at parameter t is ty = x + t². Wait, is that correct?Wait, the standard tangent equation for parametric form is ty = x + t². Let me verify.Yes, for parabola y² = 4ax, parametric equations are x = at², y = 2at. The tangent at point t is ty = x + at². Since a = 1, it's ty = x + t².So, the tangent equation is ty = x + t².Now, the point (-1, 2) lies on this tangent. So, substituting x = -1, y = 2:t*2 = -1 + t²So, 2t = t² - 1Bring all terms to one side:t² - 2t - 1 = 0Solving for t:t = [2 ± sqrt(4 + 4)] / 2 = [2 ± sqrt(8)] / 2 = [2 ± 2*sqrt(2)] / 2 = 1 ± sqrt(2)So, t1 = 1 + sqrt(2), t2 = 1 - sqrt(2)Therefore, the points of contact are:For t1: x = (1 + sqrt(2))² = 1 + 2*sqrt(2) + 2 = 3 + 2*sqrt(2), y = 2*(1 + sqrt(2)) = 2 + 2*sqrt(2). So, point P: (3 + 2*sqrt(2), 2 + 2*sqrt(2))For t2: x = (1 - sqrt(2))² = 1 - 2*sqrt(2) + 2 = 3 - 2*sqrt(2), y = 2*(1 - sqrt(2)) = 2 - 2*sqrt(2). So, point Q: (3 - 2*sqrt(2), 2 - 2*sqrt(2))Same as before. So, the points are correct.Now, the chord of contact PQ has slope 1, as calculated earlier, and passes through the focus (1,0). So, the line connecting the tangent points and the focus is PQ itself, with slope 1.Therefore, the answer is 1.I think that's solid. I've verified it through two different methods: using the slope form of the tangent and using the parametric form. Both lead to the same points of contact, and the slope of the line connecting them is 1.**Final Answer**The slope is boxed{1}.

answer：Okay, so I have this problem where I need to find the value of x such that the vectors a and b are collinear. Vector a is given as (1, -5) and vector b is (x-1, -10). Hmm, collinear vectors... I remember that means they lie on the same line, right? So, they must be scalar multiples of each other. That is, one vector can be obtained by multiplying the other by some scalar.Let me write that down. If vectors a and b are collinear, then there exists some scalar k such that:b = k * aSo, substituting the given vectors, we have:(x - 1, -10) = k * (1, -5)Which means each component of vector b is equal to k times the corresponding component of vector a. So, breaking it down into components:x - 1 = k * 1 ...(1)-10 = k * (-5) ...(2)Alright, so now I have two equations. I can solve equation (2) first to find the value of k, and then substitute that into equation (1) to find x.Starting with equation (2):-10 = k * (-5)To solve for k, I can divide both sides by -5:k = (-10) / (-5) = 2So, k is 2. Now, plug this value of k into equation (1):x - 1 = 2 * 1Simplify the right side:x - 1 = 2Now, solve for x by adding 1 to both sides:x = 2 + 1 = 3So, x should be 3. Let me double-check to make sure I didn't make any mistakes.If x is 3, then vector b becomes (3 - 1, -10) which is (2, -10). Vector a is (1, -5). Now, is (2, -10) a scalar multiple of (1, -5)? Let's see:Multiply vector a by 2: 2*(1, -5) = (2, -10). Yep, that's exactly vector b. So, that checks out.Wait, just to make sure I didn't make any calculation errors. Let me go through the steps again.Given vectors a = (1, -5) and b = (x-1, -10). For them to be collinear, the ratio of their corresponding components must be equal. So, (x - 1)/1 = (-10)/(-5). Simplifying the right side, (-10)/(-5) is 2. So, (x - 1)/1 = 2, which means x - 1 = 2, so x = 3. Yep, same result.Alternatively, I can think about the direction ratios. For two vectors to be collinear, their direction ratios must be proportional. So, the ratio of the x-components should be equal to the ratio of the y-components.So, (x - 1)/1 = (-10)/(-5). Again, (-10)/(-5) is 2, so (x - 1)/1 = 2, leading to x = 3. Same answer.I think that's solid. I don't see any mistakes in my reasoning.Another way to think about it is using the concept of slope. If two vectors are collinear, the lines they represent must have the same slope. So, the slope of vector a is (-5)/1 = -5. The slope of vector b is (-10)/(x - 1). For them to be the same, set them equal:-5 = (-10)/(x - 1)Multiply both sides by (x - 1):-5*(x - 1) = -10Divide both sides by -5:x - 1 = 2So, x = 3. Yep, same result again.I think I've approached this problem from multiple angles: using scalar multiples, direction ratios, and slope. All methods lead me to x = 3. So, I feel confident that this is the correct answer.Just to recap:1. Collinear vectors are scalar multiples of each other.2. Set up equations based on components.3. Solved for scalar k.4. Substituted back to find x.5. Verified using alternative methods to ensure correctness.Yep, that's thorough. I don't think I missed anything here.**Final Answer**The value of ( x ) is boxed{3}.

answer：To find the common prime divisors of 390 and 9450, I'll start by performing prime factorization on each number.For 390, I divide by 2 to get 195. Then, dividing by 3 gives me 65. Continuing with 5, I reach 13, which is a prime number. So, the prime factors of 390 are 2, 3, 5, and 13.Next, for 9450, I divide by 2 to get 4725. Dividing by 3 repeatedly yields 175. Dividing by 5 twice results in 7, which is prime. Therefore, the prime factors of 9450 are 2, 3, 5, and 7.Now, I'll identify the common prime factors between the two numbers, which are 2, 3, and 5.Finally, I'll calculate the sum of these common prime factors: 2 + 3 + 5 equals 10.

answer：Alright, so I've got this problem about a 3x3 matrix with 9 distinct positive numbers. Each row forms an arithmetic sequence, and the sums of each row form a geometric sequence. There are four judgments given, and I need to figure out how many of them are correct. Let me try to break this down step by step.First, let me write down what I know. The matrix is:[begin{pmatrix}a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33}end{pmatrix}]Each row is an arithmetic sequence. So, for the first row, the three numbers (a_{11}), (a_{12}), (a_{13}) form an arithmetic sequence. Similarly, the second and third rows do as well.In an arithmetic sequence, the middle term is the average of the other two. So, for the first row, (a_{12} = frac{a_{11} + a_{13}}{2}). Similarly, for the second row, (a_{22} = frac{a_{21} + a_{23}}{2}), and for the third row, (a_{32} = frac{a_{31} + a_{33}}{2}).Now, the sums of each row form a geometric sequence. So, the sum of the first row, (S_1 = a_{11} + a_{12} + a_{13}), the sum of the second row, (S_2 = a_{21} + a_{22} + a_{23}), and the sum of the third row, (S_3 = a_{31} + a_{32} + a_{33}) form a geometric sequence.In a geometric sequence, each term is a multiple of the previous one by a common ratio. So, (S_2 = S_1 cdot r) and (S_3 = S_2 cdot r = S_1 cdot r^2), where (r) is the common ratio.Since each row is an arithmetic sequence, the sum of each row can be expressed in terms of the middle term. For an arithmetic sequence of three terms, the sum is (3 times) the middle term. So, (S_1 = 3a_{12}), (S_2 = 3a_{22}), and (S_3 = 3a_{32}).Therefore, the sums (3a_{12}), (3a_{22}), (3a_{32}) form a geometric sequence. Dividing each term by 3, we see that (a_{12}), (a_{22}), (a_{32}) must also form a geometric sequence. So, that's interesting—judgment ① says that the second column must form a geometric sequence. From this reasoning, it seems that ① is correct.Moving on to judgment ②: The first column (a_{11}), (a_{21}), (a_{31}) do not necessarily form a geometric sequence. Hmm. Let me think about whether the first column has to be a geometric sequence or not.Given that each row is an arithmetic sequence, we can express each term in terms of the middle term and the common difference. Let me denote the common differences for each row as (d_1), (d_2), (d_3) respectively.So, for the first row:- (a_{11} = a_{12} - d_1)- (a_{13} = a_{12} + d_1)Similarly, for the second row:- (a_{21} = a_{22} - d_2)- (a_{23} = a_{22} + d_2)And for the third row:- (a_{31} = a_{32} - d_3)- (a_{33} = a_{32} + d_3)Now, since (a_{12}), (a_{22}), (a_{32}) form a geometric sequence, let's denote the common ratio as (r). So, (a_{22} = a_{12} cdot r) and (a_{32} = a_{22} cdot r = a_{12} cdot r^2).Now, let's look at the first column: (a_{11}), (a_{21}), (a_{31}). Let's express each term:- (a_{11} = a_{12} - d_1)- (a_{21} = a_{22} - d_2 = a_{12} r - d_2)- (a_{31} = a_{32} - d_3 = a_{12} r^2 - d_3)So, the first column is (a_{12} - d_1), (a_{12} r - d_2), (a_{12} r^2 - d_3). For this to be a geometric sequence, the ratios between consecutive terms should be constant.Let me denote the ratio as (k). Then:[frac{a_{21}}{a_{11}} = frac{a_{12} r - d_2}{a_{12} - d_1} = k]and[frac{a_{31}}{a_{21}} = frac{a_{12} r^2 - d_3}{a_{12} r - d_2} = k]So, both these ratios must equal (k). Therefore,[frac{a_{12} r - d_2}{a_{12} - d_1} = frac{a_{12} r^2 - d_3}{a_{12} r - d_2}]Cross-multiplying:[(a_{12} r - d_2)^2 = (a_{12} - d_1)(a_{12} r^2 - d_3)]This equation must hold for the first column to be a geometric sequence. However, unless the differences (d_1), (d_2), (d_3) satisfy this condition, the first column doesn't necessarily form a geometric sequence. Since the problem states that the numbers are distinct and positive, but doesn't impose any further restrictions, it's possible that (d_1), (d_2), (d_3) can be chosen such that the first column isn't a geometric sequence. Therefore, judgment ② is correct—meaning the first column doesn't necessarily form a geometric sequence.Moving on to judgment ③: (a_{12} + a_{32} > a_{21} + a_{23}). Let me see if this inequality holds.First, let's express (a_{21} + a_{23}). Since the second row is an arithmetic sequence, (a_{21} + a_{23} = 2a_{22}). So, the inequality becomes (a_{12} + a_{32} > 2a_{22}).But since (a_{12}), (a_{22}), (a_{32}) form a geometric sequence, let's denote (a_{22} = a_{12} r) and (a_{32} = a_{12} r^2). So, substituting these in:[a_{12} + a_{12} r^2 > 2 a_{12} r]Divide both sides by (a_{12}) (since all terms are positive, this is allowed):[1 + r^2 > 2r]Which simplifies to:[r^2 - 2r + 1 > 0][(r - 1)^2 > 0]Since (r) is the common ratio of a geometric sequence with distinct positive terms, (r) cannot be 1 (as that would make all terms equal, contradicting the distinctness). Therefore, ((r - 1)^2) is always positive, so the inequality holds. Therefore, judgment ③ is correct.Lastly, judgment ④: If the sum of the 9 numbers equals 9, then (a_{22} < 1). Let me analyze this.First, the total sum of all 9 numbers is the sum of the three row sums, which are (3a_{12}), (3a_{22}), (3a_{32}). So, the total sum is (3(a_{12} + a_{22} + a_{32})). We are told this equals 9, so:[3(a_{12} + a_{22} + a_{32}) = 9][a_{12} + a_{22} + a_{32} = 3]Since (a_{12}), (a_{22}), (a_{32}) form a geometric sequence, let me denote them as (a), (ar), (ar^2), where (a = a_{12}) and (r) is the common ratio.So, the sum is:[a + ar + ar^2 = 3][a(1 + r + r^2) = 3]We need to find whether (a_{22} = ar < 1).So, (ar < 1). Let's see if this must hold given the constraints.Since all terms are positive and distinct, (r neq 1). Also, since the numbers are distinct, (r) cannot be 1, and (a) must be positive.Let me consider two cases: (r > 1) and (0 < r < 1).Case 1: (r > 1). Then, (ar^2 > ar > a). So, the terms are increasing.Case 2: (0 < r < 1). Then, (ar^2 < ar < a). So, the terms are decreasing.In either case, the middle term (ar) is between (a) and (ar^2). Let's see if (ar) can be greater than or equal to 1.Suppose (ar geq 1). Then, depending on the value of (r), the sum (a(1 + r + r^2)) could be larger or smaller.But since (a(1 + r + r^2) = 3), we can express (a = frac{3}{1 + r + r^2}).So, (ar = frac{3r}{1 + r + r^2}). We need to see if (ar < 1).So, is (frac{3r}{1 + r + r^2} < 1)?Multiply both sides by (1 + r + r^2) (which is positive):[3r < 1 + r + r^2][0 < 1 - 2r + r^2][0 < (1 - r)^2]Which is always true, since a square is non-negative and (r neq 1) (as the terms are distinct). Therefore, ((1 - r)^2 > 0), so (3r < 1 + r + r^2), which implies (ar < 1). Therefore, (a_{22} = ar < 1). So, judgment ④ is correct.Wait, hold on. Let me double-check this. If (ar < 1), then (a_{22} < 1). But is this necessarily true?Wait, let me think differently. Suppose (ar = 1). Then, (a = frac{1}{r}). Then, the sum (a(1 + r + r^2) = frac{1}{r}(1 + r + r^2) = 1 + r + r^2) divided by (r). So, (1 + r + r^2) over (r) equals ( frac{1}{r} + 1 + r ). For this to equal 3:[frac{1}{r} + 1 + r = 3][frac{1}{r} + r = 2]Multiply both sides by (r):[1 + r^2 = 2r][r^2 - 2r + 1 = 0][(r - 1)^2 = 0]So, (r = 1), but this contradicts the distinctness of the terms. Therefore, (ar) cannot equal 1. Therefore, (ar) must be less than 1 because if (ar) were greater than or equal to 1, we would reach a contradiction or have equal terms. Hence, (ar < 1), so (a_{22} < 1). Therefore, judgment ④ is correct.Wait, but hold on again. Let me test with an example. Suppose (r = 2). Then, (a = frac{3}{1 + 2 + 4} = frac{3}{7}). Then, (ar = frac{3}{7} times 2 = frac{6}{7} < 1). So, that's fine.Another example: (r = frac{1}{2}). Then, (a = frac{3}{1 + frac{1}{2} + frac{1}{4}} = frac{3}{frac{7}{4}} = frac{12}{7}). Then, (ar = frac{12}{7} times frac{1}{2} = frac{6}{7} < 1). So, again, (ar < 1).Wait, but if (r) is something else, say (r = 3), then (a = frac{3}{1 + 3 + 9} = frac{3}{13}), so (ar = frac{9}{13} < 1). Similarly, for (r = frac{1}{3}), (a = frac{3}{1 + frac{1}{3} + frac{1}{9}} = frac{3}{frac{13}{9}} = frac{27}{13}), so (ar = frac{27}{13} times frac{1}{3} = frac{9}{13} < 1).Wait, so in all cases, (ar < 1). Therefore, judgment ④ is correct.Wait, but hold on another thought. Let me consider (r) approaching 0. If (r) approaches 0, then (a = frac{3}{1 + r + r^2}) approaches 3. Then, (ar) approaches 0. So, (ar) is less than 1.If (r) approaches infinity, then (a = frac{3}{1 + r + r^2}) approaches 0, so (ar) approaches 0 as well. So, in all cases, (ar) is less than 1.Therefore, judgment ④ is correct.Wait, but hold on. Let me think if there's any possible (r) where (ar) could be equal to 1 or greater than 1. From the earlier equation, when (ar = 1), we get (r = 1), which is not allowed. So, (ar) cannot be 1. And from the earlier inequality, (ar < 1). So, yes, judgment ④ is correct.Wait, but hold on another angle. Suppose (r) is negative. But wait, all numbers are positive, so (r) must be positive. So, (r > 0). Therefore, we don't have to consider negative ratios.Therefore, all four judgments seem to be correct. But wait, the problem says "the correct number of the following four judgments is ______." So, if all four are correct, the answer would be 4. But let me double-check each judgment because sometimes initial reasoning can be flawed.Re-examining judgment ①: The second column must form a geometric sequence. Since each row is arithmetic, the middle terms are (a_{12}), (a_{22}), (a_{32}), and their sums form a geometric sequence, which implies the middle terms themselves form a geometric sequence. So, yes, ① is correct.Judgment ②: The first column does not necessarily form a geometric sequence. As shown earlier, unless the differences satisfy a specific condition, the first column isn't necessarily geometric. So, ② is correct.Judgment ③: (a_{12} + a_{32} > a_{21} + a_{23}). Since (a_{21} + a_{23} = 2a_{22}), and (a_{12} + a_{32} = a + ar^2), which is greater than (2ar) because ( (r - 1)^2 > 0 ). So, ③ is correct.Judgment ④: If the total sum is 9, then (a_{22} < 1). As shown through algebra and examples, this holds true. So, ④ is correct.Wait, but the problem says "the correct number of the following four judgments is ______." So, if all four are correct, the answer is 4. But I have a feeling that maybe judgment ② isn't necessarily correct because perhaps the first column could also form a geometric sequence, but the problem says "do not necessarily form," meaning it's not always a geometric sequence, but it could be sometimes. So, the statement is correct because it's saying they don't have to, not that they can't.Wait, no, the judgment is ②: "The first column (a_{11}), (a_{21}), (a_{31}) do not necessarily form a geometric sequence." So, it's saying that it's not required, which is correct because it's possible for them to form a geometric sequence or not, depending on the differences. So, the statement is correct.Wait, but hold on. Let me think if it's possible for the first column to form a geometric sequence. Suppose the differences (d_1), (d_2), (d_3) are chosen such that (a_{11}), (a_{21}), (a_{31}) form a geometric sequence. Is that possible?Let me try to construct such a matrix.Let me choose (a_{12} = 1), (r = 2), so (a_{22} = 2), (a_{32} = 4). Then, the sums are 3, 6, 12, which is a geometric sequence with ratio 2.Now, let me choose the differences such that the first column is a geometric sequence.Let me denote the first column as (a_{11}), (a_{21}), (a_{31}). Let me assume they form a geometric sequence with ratio (k). So, (a_{21} = a_{11}k), (a_{31} = a_{11}k^2).From the arithmetic sequences in the rows:First row: (a_{11} = a_{12} - d_1 = 1 - d_1), (a_{13} = 1 + d_1).Second row: (a_{21} = 2 - d_2), (a_{23} = 2 + d_2).Third row: (a_{31} = 4 - d_3), (a_{33} = 4 + d_3).But we also have (a_{21} = a_{11}k) and (a_{31} = a_{11}k^2).So, substituting:(2 - d_2 = (1 - d_1)k)(4 - d_3 = (1 - d_1)k^2)We need to find (d_1), (d_2), (d_3), (k) such that these equations hold, and all numbers are distinct and positive.Let me choose (k = 2). Then:(2 - d_2 = (1 - d_1) times 2)(4 - d_3 = (1 - d_1) times 4)From the first equation:(2 - d_2 = 2(1 - d_1))Simplify:(2 - d_2 = 2 - 2d_1)Subtract 2 from both sides:(-d_2 = -2d_1)So, (d_2 = 2d_1)From the second equation:(4 - d_3 = 4(1 - d_1))Simplify:(4 - d_3 = 4 - 4d_1)Subtract 4:(-d_3 = -4d_1)So, (d_3 = 4d_1)Now, let's choose (d_1 = 1). Then, (d_2 = 2), (d_3 = 4).So, let's compute the matrix:First row: (a_{11} = 1 - 1 = 0), but that's not positive. So, invalid.Let me choose (d_1 = 0.5). Then, (d_2 = 1), (d_3 = 2).First row: (a_{11} = 1 - 0.5 = 0.5), (a_{13} = 1 + 0.5 = 1.5).Second row: (a_{21} = 2 - 1 = 1), (a_{23} = 2 + 1 = 3).Third row: (a_{31} = 4 - 2 = 2), (a_{33} = 4 + 2 = 6).So, the matrix is:[begin{pmatrix}0.5 & 1 & 1.5 1 & 2 & 3 2 & 4 & 6end{pmatrix}]Now, let's check the first column: 0.5, 1, 2. This is a geometric sequence with ratio 2. So, in this case, the first column does form a geometric sequence. Therefore, it is possible for the first column to form a geometric sequence. However, the judgment says "do not necessarily form a geometric sequence," which is still correct because it's not always the case—it can be sometimes. So, the judgment is correct because it's not a necessity, even though it can happen.Therefore, all four judgments are correct. But wait, the problem is from a Chinese source, and sometimes in these problems, not all judgments might hold. Let me double-check.Wait, in my example above, the first column is a geometric sequence, but the numbers are 0.5, 1, 2, which are distinct and positive. However, the second column is 1, 2, 4, which is also a geometric sequence. So, in this case, both the first and second columns are geometric sequences. But the problem only requires the sums of the rows to form a geometric sequence, and each row to be an arithmetic sequence. So, in this case, both columns are geometric, but the problem doesn't state anything about the columns, only the rows and the sums.But the judgment ② is about the first column not necessarily forming a geometric sequence. Since in some cases it can, and in others, it can't, the judgment is correct because it's not a necessity.Wait, but in my example, the first column is a geometric sequence, but the numbers in the matrix are distinct and positive. So, the first column can be a geometric sequence, but it's not required. Therefore, judgment ② is correct because it's saying it doesn't have to be, which is true.Therefore, all four judgments are correct. So, the correct number is 4.But wait, hold on. Let me think again about judgment ③: (a_{12} + a_{32} > a_{21} + a_{23}). In my example, (a_{12} = 1), (a_{32} = 4), so (1 + 4 = 5). (a_{21} + a_{23} = 1 + 3 = 4). So, 5 > 4, which holds.Another example: Let me choose (a_{12} = 2), (r = frac{1}{2}). So, (a_{22} = 1), (a_{32} = 0.5). Then, the sums are 6, 3, 1.5, which is a geometric sequence with ratio 0.5.Now, let's compute the differences.First row: (a_{11} = 2 - d_1), (a_{13} = 2 + d_1).Second row: (a_{21} = 1 - d_2), (a_{23} = 1 + d_2).Third row: (a_{31} = 0.5 - d_3), (a_{33} = 0.5 + d_3).Let me choose (d_1 = 1), so first row: 1, 2, 3.Second row: Let me choose (d_2 = 0.5), so (a_{21} = 0.5), (a_{23} = 1.5).Third row: Let me choose (d_3 = 0.25), so (a_{31} = 0.25), (a_{33} = 0.75).So, the matrix is:[begin{pmatrix}1 & 2 & 3 0.5 & 1 & 1.5 0.25 & 0.5 & 0.75end{pmatrix}]Now, let's check judgment ③: (a_{12} + a_{32} = 2 + 0.5 = 2.5). (a_{21} + a_{23} = 0.5 + 1.5 = 2). So, 2.5 > 2, which holds.Another example: Let me try with (a_{12} = 3), (r = frac{1}{3}). So, (a_{22} = 1), (a_{32} = frac{1}{3}). The sums are 9, 3, 1, which is a geometric sequence with ratio (frac{1}{3}).First row: (a_{11} = 3 - d_1), (a_{13} = 3 + d_1).Second row: (a_{21} = 1 - d_2), (a_{23} = 1 + d_2).Third row: (a_{31} = frac{1}{3} - d_3), (a_{33} = frac{1}{3} + d_3).Let me choose (d_1 = 1), so first row: 2, 3, 4.Second row: Let me choose (d_2 = 0.5), so (a_{21} = 0.5), (a_{23} = 1.5).Third row: Let me choose (d_3 = 0.1), so (a_{31} = frac{1}{3} - 0.1 approx 0.6667 - 0.1 = 0.5667), (a_{33} = frac{1}{3} + 0.1 approx 0.6667 + 0.1 = 0.7667).So, the matrix is:[begin{pmatrix}2 & 3 & 4 0.5 & 1 & 1.5 0.5667 & 0.6667 & 0.7667end{pmatrix}]Now, judgment ③: (a_{12} + a_{32} = 3 + 0.6667 approx 3.6667). (a_{21} + a_{23} = 0.5 + 1.5 = 2). So, 3.6667 > 2, which holds.Wait, but in this case, the third row's first term is approximately 0.5667, which is less than the second term of the second row, which is 1. But all numbers are distinct and positive, so that's fine.Therefore, in all these examples, judgment ③ holds. So, I think ③ is correct.Therefore, all four judgments are correct. So, the correct number is 4.But wait, the problem is presented as a multiple-choice question, but in the original problem statement, it's just asking for the number of correct judgments among four, so the answer is 4. However, sometimes in Chinese problems, the answer might be different. Let me check again.Wait, in the initial problem, it's stated that all 9 numbers are distinct positive numbers. In my first example, all numbers are distinct and positive, and all four judgments hold. In the second example, same thing. So, unless there's a case where one of the judgments fails, but I can't find such a case.Wait, let me think about judgment ② again. It says the first column does not necessarily form a geometric sequence. As shown, it can form a geometric sequence, but it's not required. So, the judgment is correct because it's not necessarily a geometric sequence—it can be, but it doesn't have to be.Therefore, all four judgments are correct, so the answer is 4.But wait, the problem is in Chinese, and sometimes the translation might be tricky. Let me make sure I understood the problem correctly.The matrix is 3x3 with 9 distinct positive numbers. Each row is an arithmetic sequence, and the sums of the rows form a geometric sequence. Then, four judgments are given about the columns and certain inequalities.Yes, that's correct. So, all four judgments are correct.Wait, but in the initial problem statement, it's written as:"In the matrix composed of 9 distinct positive numbers begin{pmatrix} left.begin{matrix}a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33}end{matrix}right.end{pmatrix}, the three numbers in each row form an arithmetic sequence, and a_{11}+a_{12}+a_{13}, a_{21}+a_{22}+a_{23}, a_{31}+a_{32}+a_{33} form a geometric sequence. The correct number of the following four judgments is ______. ① The second column a_{12}, a_{22}, a_{32} must form a geometric sequence. ② The first column a_{11}, a_{21}, a_{31} do not necessarily form a geometric sequence. ③ a_{12}+a_{32} > a_{21}+a_{23}. ④ If the sum of the 9 numbers equals 9, then a_{22} < 1."Yes, so all four judgments are correct. Therefore, the correct number is 4.But wait, I just realized that in the problem statement, it's written as "the correct number of the following four judgments is ______." So, it's expecting a number, not a box around it. But in the instructions, it says to put the final answer within boxed{}.Wait, but in the original problem, it's a fill-in-the-blank question, so the answer is 4, which is the number of correct judgments. So, I should write boxed{4}.But just to make sure, let me think if there's any case where any of the judgments might fail.Judgment ①: As the sums of the rows are geometric, and each row's sum is 3 times the middle term, so the middle terms must be geometric. So, ① is always correct.Judgment ②: The first column doesn't have to be geometric, as shown by the example where it's not. So, ② is correct.Judgment ③: As shown algebraically, (a_{12} + a_{32} > 2a_{22}), which is always true because ((r - 1)^2 > 0). So, ③ is correct.Judgment ④: As shown, when the total sum is 9, (a_{22} < 1). So, ④ is correct.Therefore, all four are correct, so the answer is 4.

answer：Okay, so I have this problem here: Given that tan α equals 2, I need to find the value of (sin α + cos α) divided by (sin α - cos α). Hmm, alright, let's see how to approach this.First off, I remember that tan α is equal to sin α over cos α. So if tan α is 2, that means sin α divided by cos α is 2. Maybe I can express sin α in terms of cos α or vice versa. Let me write that down:tan α = sin α / cos α = 2So, sin α = 2 cos α. That seems useful. Maybe I can substitute this into the expression I need to find. Let's see:(sin α + cos α) / (sin α - cos α)If I substitute sin α with 2 cos α, the numerator becomes 2 cos α + cos α, which is 3 cos α. The denominator becomes 2 cos α - cos α, which is cos α. So, putting it together:(3 cos α) / (cos α)Hmm, the cos α terms cancel out, leaving me with 3. Wait, is that it? That seems too straightforward. Maybe I should double-check.Alternatively, I could use the identity that relates tan α to sin α and cos α. Since tan α is 2, I can imagine a right triangle where the opposite side is 2 and the adjacent side is 1. Then, the hypotenuse would be sqrt(1^2 + 2^2) = sqrt(1 + 4) = sqrt(5). So, in this triangle, sin α is opposite over hypotenuse, which is 2/sqrt(5), and cos α is adjacent over hypotenuse, which is 1/sqrt(5). Let me plug these into the expression:(2/sqrt(5) + 1/sqrt(5)) / (2/sqrt(5) - 1/sqrt(5))Combine the terms in the numerator and denominator:Numerator: (2 + 1)/sqrt(5) = 3/sqrt(5)Denominator: (2 - 1)/sqrt(5) = 1/sqrt(5)So, the expression becomes (3/sqrt(5)) / (1/sqrt(5)). The sqrt(5) terms cancel out, leaving 3/1, which is 3. Okay, so that confirms it.Wait, but I remember sometimes when dealing with trigonometric identities, especially when dividing, you have to be careful about the signs. Since tan α is positive, α could be in either the first or third quadrant. But in the first quadrant, both sin and cos are positive, while in the third quadrant, both are negative. Let me check if that affects the result.If α is in the first quadrant, sin α and cos α are positive, so the expression is (positive + positive)/(positive - positive). If α is in the third quadrant, sin α and cos α are negative, so the expression becomes (negative + negative)/(negative - negative). Let's compute both cases.Case 1: First quadrant.sin α = 2/sqrt(5), cos α = 1/sqrt(5)(2/sqrt(5) + 1/sqrt(5)) / (2/sqrt(5) - 1/sqrt(5)) = (3/sqrt(5)) / (1/sqrt(5)) = 3Case 2: Third quadrant.sin α = -2/sqrt(5), cos α = -1/sqrt(5)(-2/sqrt(5) + (-1)/sqrt(5)) / (-2/sqrt(5) - (-1)/sqrt(5)) = (-3/sqrt(5)) / (-1/sqrt(5)) = (-3)/(-1) = 3So, in both cases, the result is 3. That's good to know. So regardless of the quadrant, the value is 3.Alternatively, another method I can think of is using the formula for (sin α + cos α)/(sin α - cos α). Maybe I can manipulate this expression by dividing numerator and denominator by cos α. Let's try that.Divide numerator and denominator by cos α:(sin α / cos α + cos α / cos α) / (sin α / cos α - cos α / cos α)Simplify each term:(tan α + 1) / (tan α - 1)Since tan α is given as 2, substitute that in:(2 + 1) / (2 - 1) = 3 / 1 = 3Wow, that was even quicker. So, that's another way to approach it without dealing with the hypotenuse or the triangle. Just by dividing numerator and denominator by cos α, I can express everything in terms of tan α, which is given. That seems like a more efficient method.I think I've got it. The answer is 3. But just to make sure, let me recap the steps:1. Recognize that tan α = 2 implies sin α = 2 cos α.2. Substitute sin α in the expression (sin α + cos α)/(sin α - cos α) to get (2 cos α + cos α)/(2 cos α - cos α) = 3 cos α / cos α = 3.3. Alternatively, divide numerator and denominator by cos α to express the ratio in terms of tan α, which is 2, leading to (2 + 1)/(2 - 1) = 3.4. Consider the possibility of α being in different quadrants, but realize that regardless of the sign, the ratio remains 3.So, all methods lead to the same result, which is 3. I feel confident about this answer now.**Final Answer**The value of the expression is boxed{3}.